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Signature-permutation of a Catalan automorphism: Reflect a rooted plane binary tree; Deutsch's 1998 involution on Dyck paths.
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%I #36 Jan 14 2024 08:58:50

%S 0,1,3,2,8,7,6,5,4,22,21,20,18,17,19,16,15,13,12,14,11,10,9,64,63,62,

%T 59,58,61,57,55,50,49,54,48,46,45,60,56,53,47,44,52,43,41,36,35,40,34,

%U 32,31,51,42,39,33,30,38,29,27,26,37,28,25,24,23,196,195,194,190,189

%N Signature-permutation of a Catalan automorphism: Reflect a rooted plane binary tree; Deutsch's 1998 involution on Dyck paths.

%C Deutsch shows in his 1999 paper that this automorphism maps the number of doublerises of Dyck paths to number of valleys and height of the first peak to the number of returns, i.e., that A126306(n) = A127284(a(n)) and A126307(n) = A057515(a(n)) hold for all n.

%C The A000108(n-2) n-gon triangularizations can be reflected over n axes of symmetry, which all can be generated by appropriate compositions of the permutations A057161/A057162 and A057163.

%C Composition with A057164 gives signature permutation for Donaghey's Map M (A057505/A057506). Embeds into itself in scale n:2n+1 as a(n) = A083928(a(A080298(n))). A127302(a(n)) = A127302(n) and A057123(A057163(n)) = A057164(A057123(n)) hold for all n.

%H JungHwan Min, <a href="/A057163/b057163.txt">Table of n, a(n) for n = 0..10000</a>

%H Emeric Deutsch, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00370-7">An involution on Dyck paths and its consequences</a>, Discrete Math., 204 (1999), no. 1-3, 163-166.

%H Indranil Ghosh, <a href="/A057163/a057163.txt">Python program for computing this sequence, translated from Maple code</a>.

%H Antti Karttunen, <a href="/A089408/a089408.c.txt">C program which computes this sequence</a>.

%H Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, <a href="https://arxiv.org/abs/2012.04625">Finding structure in sequences of real numbers via graph theory: a problem list</a>, arXiv:2012.04625, Dec 08, 2020.

%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>

%F a(n) = A083927(A057164(A057123(n))).

%e This involution (self-inverse permutation) of natural numbers is induced when we reflect the rooted plane binary trees encoded by A014486. E.g., we have A014486(5) = 44 (101100 in binary), A014486(7) = 52 (110100 in binary) and these encode the following rooted plane binary trees, which are reflections of each other:

%e 0 0 0 0

%e \ / \ /

%e 1 0 0 1

%e \ / \ /

%e 0 1 1 0

%e \ / \ /

%e 1 1

%e thus a(5)=7 and a(7)=5.

%p a(n) = A080300(ReflectBinTree(A014486(n)))

%p ReflectBinTree := n -> ReflectBinTree2(n)/2; ReflectBinTree2 := n -> (`if`((0 = n),n,ReflectBinTreeAux(A030101(n))));

%p ReflectBinTreeAux := proc(n) local a,b; a := ReflectBinTree2(BinTreeLeftBranch(n)); b := ReflectBinTree2(BinTreeRightBranch(n)); RETURN((2^(A070939(b)+A070939(a))) + (b * (2^(A070939(a)))) + a); end;

%p NextSubBinTree := proc(nn) local n,z,c; n := nn; c := 0; z := 0; while(c < 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); od; RETURN(z); end;

%p BinTreeLeftBranch := n -> NextSubBinTree(floor(n/2));

%p BinTreeRightBranch := n -> NextSubBinTree(floor(n/(2^(1+A070939(BinTreeLeftBranch(n))))));

%t A014486Q[0] = True; A014486Q[n_] := Catch[Fold[If[# < 0, Throw[False], If[#2 == 0, # - 1, # + 1]] &, 0, IntegerDigits[n, 2]] == 0]; tree[n_] := Block[{func, num = Append[IntegerDigits[n, 2], 0]}, func := If[num[[1]] == 0, num = Drop[num, 1]; 0, num = Drop[num, 1]; 1[func, func]]; func]; A057163L[n_] := Function[x, FirstPosition[x, FromDigits[Most@Cases[tree[#] /. 1 -> Reverse@*1, 0 | 1, All, Heads -> True], 2]][[1]] - 1 & /@ x][Select[Range[0, 2^n], A014486Q]]; A057163L[11] (* _JungHwan Min_, Dec 11 2016 *)

%o (Scheme implementations of this automorphism that acts on S-expressions, i.e., list-structures:)

%o (CONSTRUCTIVE IMPLEMENTATION:) (define (*A057163 s) (cond ((not (pair? s)) s) (else (cons (*A057163 (cdr s)) (*A057163 (car s))))))

%o (DESTRUCTIVE IMPLEMENTATION:) (define (*A057163! s) (cond ((pair? s) (*A069770! s) (*A057163! (car s)) (*A057163! (cdr s)))) s)

%Y This automorphism conjugates between the car/cdr-flipped variants of other automorphisms, e.g., A057162(n) = a(A057161(a(n))), A069768(n) = a(A069767(a(n))), A069769(n) = a(A057508(a(n))), A069773(n) = a(A057501(a(n))), A069774(n) = a(A057502(a(n))), A069775(n) = a(A057509(a(n))), A069776(n) = a(A057510(a(n))), A069787(n) = a(A057164(a(n))).

%Y Row 1 of tables A122201 and A122202, that is, obtained with FORK (and KROF) transformation from even simpler automorphism *A069770. Cf. A122351.

%K nonn

%O 0,3

%A _Antti Karttunen_, Aug 18 2000

%E Equivalence with Deutsch's 1998 involution realized Dec 15 2006 and entry edited accordingly by _Antti Karttunen_, Jan 16 2007